systems biology samsi opening workshop algebraic methods in systems biology and statistics september...
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Systems biology
SAMSI Opening WorkshopAlgebraic Methods in Systems Biology and Statistics
September 14, 2008
Reinhard LaubenbacherVirginia Bioinformatics Institute
and Mathematics DepartmentVirginia Tech
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“Living systems, being nonlinear dynamical systems, have properties different from their constituents in isolation, properties which emerge from the interactions among the molecular constituents; accordingly, it is the organization of these intermolecular processes in organisms that underlies their characteristic living properties. A reductionist or antireductionist strategy alone does not do justice to this claim. A new strategy seems needed […]
F. C. Boogerd et al., 2007
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Genomics/proteomics
Interactions between moleculesIntracellular networks
Tissue level processescomplexity
Whole organism
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Y. Lazebnik, Cancer Cell, 2002
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G. Koh et al., Bioinformatics, 2006
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Model Types
Ideker, Lauffenburger, Trends in Biotech 21, 2003
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Discrete models of molecular networks
“[The] transcriptional control of a gene can be described by a discrete-valued function of several discrete-valued variables.”
“A regulatory network, consisting of many interacting genes and transcription factors, can be described as a collection of interrelated discrete functionsand depicted by a wiring diagram similar to the diagram of a digital logic circuit.”
R. Karp, 2002
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Nature 406 2000
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Discrete modeling frameworks
1. Boolean networks and cellular automata (including probabilistic and sequential BNs)
2. Polynomial dynamical systems over finite fields
3. Logical models
4. Dynamic Bayesian networks
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Boolean networks
Definition. Let f1,…,fn be Boolean functions in variables x1,…,xn. A Boolean network is a time-discrete dynamical system
f = (f1,…,fn) : {0, 1}n → {0, 1}n
The state space of f is the directed graph with the elements of {0,1}n as nodes. There is a directed edge b → c iff f(b) = c.
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f1 = NOT x2
f2 = x4 OR (x1 AND x3)
f3 = x4 AND x2
f4 = x2 OR x3
Boolean networks
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The phase plane
Compound
y
Compound x
dx /dt = f (x,y)dy /dt = g(x,y)
(xo ,yo)
dx = f (xo ,yo) dt
dy = g(xo ,yo) dt
Courtesy J. Tyson
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Boolean network models in biology
Stuart A. Kauffman
Metabolic stability and epigenesis in randomly constructed genetic nets
J. Theor. Biol. 22 (1969) 437-467.
Boolean networks as models for genetic regulatory networks:
Nodes = genes, functions = gene regulation
Variable states: 1 = ON, 0 = OFF
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Polynomial dynamical systems
Note: {0, 1} = k has a field structure (1+1=0).
Fact: Any Boolean function in n variables can be expressed uniquely as a polynomial function in
k[x1,…,xn] / <xi2 – xi>,
and conversely.
Proof: x AND y = xyx OR y = x+y+xy
NOT x = x+1(x XOR y = x+y)
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Polynomial dynamical systems
Let k be a finite field and f1, … , fn k[x1,…,xn]
f = (f1, … , fn) : kn → kn
is an n-dimensional polynomial dynamical system over k.
Natural generalization of Boolean networks.
Fact: Every function kn → k can be represented by a polynomial, so all finite dynamical systems kn → kn
are polynomial dynamical systems.
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Example
k = F3 = {0, 1, 2}, n = 3
f1 = x1x22+x3,
f2 = x2+x3,
f3 = x12+x2
2.
Dependency graph(wiring diagram)
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Sequential polynomial systems
k = F3 = {0, 1, 2}, n = 3
f1 = x1x22+x3
f2 = x2+x3
f3 = x12+x2
2
σ = (2 3 1) update schedule:
First update f2.
Then f3, using the new value of x2.
Then f1, using the new values of x2 and x3.
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Sequential systems as biological models
• Different regulatory processes happen on different time scales
• Stochastic effects in the cell affect the “update order” of variables representing different chemical compounds at any given time
Therefore, sequential update in models of regulatory networks adds realistic feature.
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Stochastic models
Polynomial dynamical systems can be modified:
• Choose random update order for each update
(see Sontag et al. for Boolean case)
• Choose an update function at random from a collection at each update
(see Shmulevich et al. for Boolean case)
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Logical models
E. Snoussi and R. ThomasLogical identification of all steady states: the concept of feedback loop characteristic statesBull. Math. Biol. 55 (1993) 973-991
Key model features: • Time delays of different lengths for different
variables are important• Positive and negative feedback loops are important
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Model description
Basic structure of logical models:
1. Sets of variables x1, … , xn; X1, … , Xn
(Xi = genes and xi = gene products, e.g., proteins. A gene product x regulates a gene Y, with a certain time delay.)
Each variable pair xi, Xi takes on a finite number of distinct states or thresholds (possibly different for different i), corresponding to different modes of action of the variables for different concentration levels.
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Model description (cont.)
2. A directed weighted graph with the xi as nodes and threshold levels, indicating regulatory relationships and at what levels they occur.
Each edge has a sign, indicating activation (+) or inhibition (-).
3. A collection of “logical parameters” which can be used to determine the state transition of a given node for a given configuration of inputs.
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Features of logical models
• Sophisticated models that include many features of real networks
• Ability to construct continuous models based on the logical model specification
• Models encode intuitive network properties
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An Example
X = z
Y = x
Z = y
xy
z
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Features of logical models
• Include many features of real biological networks
• Intuitive but complicated formalism and model description
• Difficult to study as a mathematical object
• Difficult to study dynamics for larger models
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Equivalence of models
Theorem. (A. Veliz-Cuba, A. Jarrah, L.) A logical model can be encoded as a PDS, without loss of information.
(Boolean case: H. Siebert)
(Similarly, certain types of Petri nets can be encoded as PDS.)
This aids model analysis.
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Dynamic Bayesian networks
Definition. A Bayesian network (BN) is a representation of a joint probability distribution over a set X1, … , Xn of random variables. It consists of
• an acyclic graph with the Xi as vertices. A directed edge indicates a conditional dependence relation
• a family of conditional distributions for each variable, given its parents in the graph
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BN models of gene regulatory networks
Can use BNs to model gene regulatory networks:
Random variables Xi ↔ genes
Directed edges ↔ regulatory relationships
Problem: BNs cannot have directed loops. Hence cannot model feedback loops.
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Dynamic Bayesian networks
Definition. A dynamic Bayesian network (DBN) is a representation of the stochastic evolution of a set of random variables {Xi}, using discrete time.
It has two components:• a directed graph (V, E) encoding conditional
dependence conditions (as before);• a family of conditional probability distributions
P(Xi(t) | Pai(t-1)), where Pai = {Xj | (Xj, Xi) E}
(Doyer et al., BMC Bioinformatics 7 (2006) )
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Dynamic Bayesian networks
DBNs generalize Hidden Markov Models.
Recently used for inference of gene regulatory networks from time courses of microarray data.
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Open problems
• Find good model inference methods (system identification) using “omics” data
• Find experimental design strategies appropriate for systems biology
• Formalize systems biology along the lines of mathematical systems theory